A classical theorem due to Young states that the cosine polynomial

$$ C_n(x)=1+\sum_{k=1}^{n}\frac{\cos(kx)}{k} $$

is positive for all $n\geq1$ and $x\in(0,\pi)$. We prove the following refinement. For all $n\geq2$ and $x\in[0,\pi]$ we have

$$ \tfrac{1}{6}+c(\pi-x)^2\leq C_n(x), $$

with the best possible constant factor

$$ c=\min_{0\leq t\lt\pi}\frac{5+6\cos(t)+3\cos(2t)}{6(\pi-t)^2}=0.069\dots. $$

We study sets of range uniqueness (SRUs) for analytic functions inside a disc of an algebraically closed field $K$ complete with respect to an ultrametric absolute value. The SRUs we obtain are converging sequences. We first obtain results that look like those known in $\mathbb{C}$ but involve a weaker hypothesis than in $\mathbb{C}$: let $(a_n)$ be a sequence of limit $a$ in a disc $d(a,r^-)$ such that $|a_n-a|$ is a strictly decreasing sequence. If the sequence $(a_n)$ does not make an SRU for the set $\mathcal{A}(d(a,r^-))$ of analytic functions inside $d(a,r^-)$, then, for a certain integer $k\in\mathbb{Z}$, the sequence

$$ \bigg(\frac{a_{n+k}-a}{a_n-a}\bigg) $$

has a finite limit in $K$ and the sequence

$$ \bigg(\frac{\log|a_{n+k}-a|}{\log|a_n-a|}\bigg) $$

has a finite rational limit. Next, we show that if the sequence

$$ \frac{\log(a_{n+1}-a)}{\log(a_n-a)} $$

converges to a limit $b\geq1$ in such a way that $-b\log|a_{n}-a|\lt-b\log|a_{n+1}-a|$ and if $\log|a_{n}-a|-b\log|a_{n+1}-a|$ has limit $0$ or $+\infty$ and if $b^k\notin\mathbb{Q}$ whenever $b>1$ and $k\in \mathbb{N}^*$, then the sequence $(a_n)$ is an SRU for $\mathcal{A}(d(a,r^-))$. In particular, for every $\gamma\in\;]0,1[\;\cup\;]1,+\infty[$, $L\in\mathbb{Q}\;\cap\;]0,+\infty[$ and $b\geq 1$, there exist SRUs for $\mathcal{A}(d(a,r^-))$ of the form $\{a_n\mid n\in\mathbb{N}\}$ such that

$$ \lim_{n\rightarrow+\infty}\frac{-\log|a_n-a|}{b^nn^\gamma}=L. $$

For example, if $\gamma\in\mathbb{N}$ with $\gamma\neq0,1$, there exist SRUs of the form $\{a_n\mid n\in\mathbb{N}\}$ such that $-\log |a_n-a|=Ln^\gamma$ for all $n\in\mathbb{N}^*$. The latter result ceases to hold when $\gamma=1$. Many examples and counterexamples are provided.

We define a partial Radon transform mapping functions on $\mathbb{R}^{n+l}$ to functions on $\mathbb{R}^{n}$ which intertwines the Laplace operator on the two spaces. As a consequence, transplantation formulae relating the radial eigenfunctions of the Laplacian on Euclidean spaces of different dimensions are obtained. Our formulae provide a geometric interpretation of integral formulae for Bessel functions of Abel type, which are found useful in potential theory. The formulae portray a view of Hadamard's method of descent within the realm of harmonic analysis, allowing the transplant of local problems from even dimensions to odd dimensions and unifying the techniques of several authors.

The principal results in this paper are concerned with the description of domains of infinitesimal generators of strongly continuous groups of isometries in non-commutative operator spaces $E(\mathcal{M},\tau)$, which are induced by $\mathbb{R}$-flows on $\mathcal{M}$. In particular, we are concerned with the description of operator functions which leave the domain of such generators invariant in all symmetric operator spaces, associated with a semi-finite von Neumann algebra $\mathcal{M}$ and a separable function space $E$ on $(0,\infty)$. Furthermore, we apply our results to the study of operator functions for which $[D,x]\in E(\mathcal{M},\tau)$ implies that $[D,f(x)]\in E(\mathcal{M},\tau)$, where $D$ is an unbounded self-adjoint operator. Our methods are partly based on the recently developed theory of double operator integrals in symmetric operator spaces and the theory of adjoint $C_{0}$-semigroups.

CoGalois groups appear in a natural way in the study of covers. They generalize the well-known group of covering automorphisms associated with universal covering spaces. Recently, it has been proved that each quasi-coherent sheaf over the projective line $\bm{P}^1(R)$ ($R$ is a commutative ring) admits a flat cover and so we have the associated coGalois group of the cover. In general the problem of computing coGalois groups is difficult. We study a wide class of quasi-coherent sheaves whose associated coGalois groups admit a very accurate description in terms of topological properties. This class includes finitely generated and cogenerated sheaves and therefore, in particular, vector bundles.

If $A$ is a stable basis algebra of rank $n$, then the set $S_{n-1}$ of endomorphisms of rank at most $n-1$ is a subsemigroup of the endomorphism monoid of $A$. This paper gives a number of necessary and sufficient conditions for $S_{n-1}$ to be generated by idempotents. These conditions are satisfied by finitely generated free modules over Euclidean domains and by free left $T$-sets of finite rank, where $T$ is cancellative monoid in which every finitely generated left ideal is principal.

For a large integer $m$, we obtain asymptotic formulae for the number of solutions of certain congruences modulo $m$ with several variables, where the variables belong to special sets of residue classes modulo $m$. In particular, we obtain new information on the exceptional set of the multiplication table problem in the residue ring modulo $m$.

Given two intersecting domains, we investigate the boundary behaviour of the quotient of Martin kernels of each domain. To this end, we give a characterization of minimal thinness for a difference of two subdomains in terms of Martin kernels of each domain. As a consequence of our main theorem, we obtain the boundary growth of the Martin kernel of a Lipschitz domain, which corresponds to earlier results for the boundary decay of the Green function for a Lipschitz domain investigated by Burdzy, Carroll and Gardiner.

The conditions under which the solution of the non-local thermistor problem

\begin{gather*} u_t=\Delta u+\frac{\lambda f(u)}{(\int_{\varOmega}f(u)\,\mathrm{d}x)^{2}},\quad x\in\varOmega\subset\mathbb{R}^N,\ N\geq2,\ t>0, \\ \frac{\partial u(x,t)}{\partial\nu}+\beta(x)u(x,t)=0,\quad x\in\partial\varOmega,\ t>0, \\ u(x,0)=u_0(x),\quad x\in\varOmega, \end{gather*}

blows up are investigated. We assume that $f(s)$ is a decreasing function and that it is integrable in $(0,\infty)$. Considering a suitable functional we prove that for all $\lambda\gt0$ the solution of the Neumann problem blows up in finite time. The same result is obtained for the Robin problem under the assumption that $\lambda$ is sufficiently large $(\lambda\gg 1)$. In the proof of existence of blow-up for the Dirichlet problem we use the subsolution technique. We are able to construct a blowing-up lower solution under the assumption that either $\lambda\gt\lambda^*$ or $0\lt\lambda\lt\lambda^*$, for some critical value $\lambda^*$, and that the initial condition is sufficiently large provided also that $f(s)$ satisfies the decay condition $\int_0^\infty[sf(s)-s^2f'(s)]\,\mathrm{d} s\lt\infty$.

This paper computes the Box and Hausdorff dimensions of the intersections of the Sierpinski carpet with planar lines of rational slopes.

We primarily consider here the $L^2$ mapping properties of a class of strongly singular Radon transforms on the Heisenberg group $\mathbb{H}^n$; these are convolution operators on $\mathbb{H}^n$ with kernels of the form $M(z,t)=K(z)\delta_0(t)$, where $K$ is a strongly singular kernel on $\mathbb{C}^n$. Our results are obtained by using the group Fourier transform and uniform asymptotic forms for Laguerre functions due to Erdélyi.

We also discuss the behaviour of related twisted strongly singular operators on $L^2(\mathbb{C}^n)$ and obtain results in this context independently of group Fourier transform methods. Key to this argument is a generalization of the results for classical strongly singular integrals on $L^2(\mathbb{R}^d)$.

In this paper we introduce the specialization of an $S$-module which is a direct limit of a direct system of finitely generated $S$-modules indexed by $\mathbb{N}$. This specialization preserves the Buchsbaum property, the generalized Cohen–Macaulay property and the Castelnuovo–Mumford regularity of a module.

Maximal regularity for an integro-differential equation with infinite delay on periodic vector-valued Besov spaces is studied. We use Fourier multipliers techniques to characterize periodic solutions solely in terms of spectral properties on the data. We study a resonance case obtaining a compatibility condition which is necessary and sufficient for the existence of periodic solutions.

The purpose of this paper is to study the influence of $c$-supplemented minimal subgroups on the $p$-nilpotency of finite groups. We obtain ‘iff' and ‘localized' versions of theorems of Itô and Buckley on nilpotence, $p$-nilpotence and supersolvability.